The impossible geometry problem that will test your brain.

We can do that together. Welcome back to my channel. A lot of people won't be able to do it. This, but let's go on and do that first. Above all, we know that this shaded portion is a square. The area of a square is length squared, so let's call the length of this square x, okay. so it means that this area is equal to x squared, that is the first thing; now I want us to Complete this as a right angle, okay? So let's conclude this. We have, okay, now we have completed. This is a right angle and this is a right angle; if we call this point F right now, recall that from e to d is three; if we call e f i and call d f j, this right now implies that using Pythagorean theorem in triangle EDF: in triangle DEF, you can see that i squared I square it; let's leave this x here. i squared plus j squared equals three squared. and we call this equation one; now this is what using the Pythagorean theorem gives us, and we're going to Leave that okay. Now, once again, let's go ahead and join from here to here—we are creating another. right-angled triangle, so from here to here, right? Okay, so this is the right angle. If this is the first When you come across this lovely channel, please hit the subscription button and turn on Notification bell so you don't miss our videos. Okay, we upload our videos as often as possible. Tune in whenever it's convenient for you—there must be something new for you to watch. Give us a thumbs up. Up if you love what we're doing, don't forget to drop your comments—we're always in the comments. section reading your suggestions and your texts, all right, thank you very much. Now, if you look if we say that from here to here is x, then that means this is also x, so if we Call this g, then f g is also x because from here to here is equal, right? Now from here to here is also equal to here to here, so df is also equal to cg, which is equal to j, so we can write cg to be j now also in triangle in triangle e c e g, so in triangle c e g using the Pythagorean theorem We can also see that i plus x squared, so i plus x all squared, okay, plus j squared plus j squared. will be equal to five squared times ec squared, five squared. Now let's call this Equation Two. recall that a plus b all squared is equal to a squared plus 2ab plus b squared, okay, so if we We are going to have i squared plus 2ix plus x squared. plus x squared then plus j squared equals five squared, okay, but recall, recall that i squared plus j squared according to equation one is three squared, okay, so we are going to Replace i squared plus j squared with three squared, so now we have three squared plus 2ix. plus x squared equals 25; 5 squared is 25, right? So now we have 9 plus 2ix plus x squared. is 25 now if we subtract 9 from both sides we will have 2ix plus x squared is equal to 25 minus 9 is 16, is 16, okay? So, permit me to erase this side of the board. Okay, let me erase this side. Again, but before I do that, let me write out my first equation: i squared plus j squared. is three squared okay all right now have you subscribed to my channel if you have not then okay, so this is where we are, so this is where we are: that 2ix plus x squared is equal to 16 Now let's move x squared to this side, and we are going to have 2x is equal to 16 minus x squared. Now I want to make x the subject of the formula. So I will divide both sides by 2x now; this will remove this, this will remove this. Now we have i equals 16 minus x squared divided by 2x. Now I'm going to write this. somewhere here because it's important to me, so I'll have i equal to 16 minus x squared divided by 2x, okay. Now we need to do something else. Are you ready? Are you ready? Okay, now if You look at this board now; if we decide to complete this triangle here, okay, if we decide to do so, then... To complete the triangle, we are going to have something like this—this is six. Now let's complete it. So we have this, and we have this. Okay, did you notice anything? Yes, from here to here is I, so from Here to here is also i, and we have two parts of this side; from here to here is x, and from here to here is... Here is J. Please pardon my board. Okay, now if we use, uh, in triangle, let's call this hash, all right. So in triangle A A' A'', you will notice that i squared plus x plus j squared, right? x plus j squared will be equal to what from here to here a e squared, which is six is equal to six squared, all right? Now I want to make sure my mic is still on. Okay, it's still on. Now, if you look at this, we can now solve that i squared plus, now we expand this as x squared. plus 2 j x plus j squared equals 36; now remember that i squared plus j squared equals 3 square right, I squared plus J squared is 3 squared, so we can now replace this and this with 3 squared, so now we have 3 squared plus x squared plus 2jx equal to 36 So we have 9 plus x squared plus 2 j x equals 36. Now let's subtract 9 from both sides. and we will have x squared plus 2j x plus 27 okay is equal to 27 all right now let's transfer Moving x squared to the other side, we have 2jx equal to 27 minus x squared. Now I want to make j is the subject of the formula, so I'm going to divide both sides by 2 over x, divide by 2x. Divide by 2x, so this and this, we go, this and this, we go, and we have another amazing Um, let me remove this because we've already finished using it. We've finished using it, okay? Alright, so now I have another equation in terms of what j. So my j now will be equal to 27 minus x squared, 27 minus x squared divided by 2x, divided by 2x, I basically wrote it Now recall that i squared plus j squared is 3 squared, right? So i squared plus j squared is equal to 9; now remember that our i is 16 minus x squared divided by 2x, so we have 16 minus x squared divided by 2x all squared plus our j squared is 27 minus x squared over 2x 27 minus x squared over 2x all squared equals 9. Now, if we expand this, we are going to have (16 minus x squared) squared over 2 squared is 4, and x squared is x squared, okay, plus the The same thing here: (27 minus x squared) all squared over 4 x squared equals 9. Now let's multiply. divided by 4x squared, so multiply both sides by 4x squared. Now you're going to have 16 minus x The square of (x squared plus 27 minus x squared) squared equals 9 multiplied by 4x squared. So if we solve further, we are going to have 16. Now there is something I noticed: we have x squared. here x squared here x squared here now to make this to become simpler to solve we can say let x squared equals a, all right. Now at this point, I think it's high time I erase this because We have finished using the diagram. Okay, now let's proceed. Let's proceed. We said let's x. be equal to a, so we can now rewrite this as (16 minus a) squared plus (27 minus a) squared is equal to 9 times 4, which is 36, and x squared is a. All right, now let's open this bracket. 16 squared gives us 256 minus 2 times 16, which is 32, so we have 32a plus a squared plus Now 27 squared gives us 729. Um, 2 times 27 is 54, so we have minus 54. a and then minus plus a squared, and everything is equal to 36a. Now let us organize this. a squared plus a squared gives us 2a squared minus 32a minus 54a gives us minus 86a then minus 36a, right? That gives us minus 122. So we have minus 122a now for our We have 729 plus 256, so 256 plus 729 gives us 985, so we have plus 985 and everything is now equal to zero, so we can now solve this using the quadratic equation formula, so using quadratic equation formula a will be equal to minus minus 1 2 2 right plus or minus square -4 multiplied by 2 multiplied by 985 we are using the quadratic equation formula Okay, now everything divided by 2 multiplied by 2, so we have a will be minus minus, which is plus. so we have 122 plus or minus the square root of now 122 squared will give us 14,884, so we have 14,884, then minus, we have 4 times 2 times 985, and that is 7,880. everything divided by 4, I hope I won't make a mistake here, so we have a is equal to 122 plus or minus the square root of 14,884 minus 7,880, and that will give us seven thousand and four everything divided by four, all right. Moving forward, we have a is equal to 1 2 2. plus or minus the square root of seven thousand and four is approximately 83.6 or 3.7 It's 3.7 now, everything divided by 4, therefore we have two values of a, right? We have two values. a could be 122 plus 83.7 divided by 4, or a is equal to 122 plus 83.7, okay, minus this time minus 3.7 divided by 4; so if we solve this, we are going to have 122 plus 83.7, which is 205.7 205.7 divided by 4 is 51.4, so we have 51.4 for a. All right, now a could be 122 minus 83.7, that is 38.3. Now, 38.3 divided by 4 gives us approximately 9.6, so now we Have two values of a, and remember when we said let x squared be equal to a, and remember that x is the side of our square, which means this is our area, because the area of a square is x squared. Right, so it means that our area could have two possible values: our area is 51.4. square units or 9.6 square units, but the truth is that neither of them can be the area One has to go, and how do we determine which one is right and which one is not right? If you're still watching all the way to here, that means you're truly a very dedicated fan of JJ. Thank you. So much for the support. Now let's go ahead. If you look at this, we have something that we can use to Prove the one that is right and the one that is wrong. Now remember where we said, let r i remember. Where are we? We got it as 16 minus x squared divided by 2x. Please, you can go back in the video. and you are going to see that so i equals 16 minus x squared divided by 2x, you can also use j to Check this now, remembering that I is part of our words diagram and it is something that is physical. It can never be negative, right? So we have a positive value here. If we have a positive value, It simply means that our x squared must be less than or equal to 16 for this to remain positive. Our x squared must be less than or equal to 16. Now let's check the two values that we got. you can see that from the first value we have 51.4, which is greater than 16, so this automatically This disqualifies it from being the answer, and now we have 9.6, which is definitely less than 16. Therefore, in conclusion, we can say that the area of the square is 9.6 units squared. Thank you so Thanks so much for watching, and see you in my next video, bye!